In Abramenko-Brown's book "Building Theory and Applications" section 3.4.1 page 129, the authors define the 'opposite' of a folding of a thin chamber complex. They mention that the opposite is defined only on the chambers and therefore it is not necessarily a folding itself (i.e. a simplicial map). A folding is called reversible if its opposite is indeed a folding. In Garrett's book the same subtlety appears.
Question : I am looking for an example of a non reversible folding of a thin chamber complex. The example should not be a Coxeter complex where the foldings are always reversible, I believe.
Start with a regular triangulation of the plane, say the one by equilateral triangles. Note that this is a thin chamber complex. Moreover, you can reflect through any of the lines in the triangulation. Now take two vertices which are of distance at least three apart and glue them together. The result is still a thin chamber complex. (I needed the vertices to be sufficiently far apart for the result to still be a simplicial complex.) Now fix a line which has both identified vertices to the same side of it (call this half-space with two points identified $\Phi$). Then one can fold across the line in one direction, toward $\Phi$, by first identifying two vertices on one side and then folding. You can't fold in the other direction because the fold was not a bijection on vertices. (To go the other way, you'd have to disidentify two vertices, which is not even going to give you a function.) This fold works fine on chambers and does everything you want it to on codimension 1 faces. It messes up at codimension 2.